A version of Theorem REF  was obtained earlier by Hodgson and Rivin [1]} for convex polyhedra, following work of Andreev [2]}. Note that taken together, the condition that \(K<1\)  and that closed geodesics have length \(L>2\pi \)  is (almost) equivalent to asking that the metric is (globally) “strongly” \(CAT(1)\) , and it is this strong \(CAT(1)\)  condition that we will use below for simplicity.
